They were probably my first 'favourite band', getting hooked on them when I was a kid.
For many years they were also, along with the likes of Pixies, The Breeders and Dinosaur Jr, a proper 'Indie' band; that is, a band signed to an independent record label, as opposed to the crappy pop bands of the 2000s. I had to stop saying that I liked some Indie bands for fear of people thinking that I listened to Razorlight.
After New Adventure In Hi-Fi I lost interest, what with Up being bloody awful. They've had a few decent songs here and there since but they've basically just become a tribute act to their former selves.